Download Hypothesis 1.key

Overview
The Basics of a
Hypothesis Test
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Alternative way to make inferences from a sample to
the Population is via a Hypothesis Test
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A hypothesis test is based upon
Dr Tom Ilvento
Department of Food and Resource Economics
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Newspaper Print Problem
(BLACKNESS.xls)
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A point estimate from a sample
Knowledge of a sampling distribution
A Null hypothesis
A probability level of being wrong that we are
comfortable with, called !
All via a rare event approach – we want to see how
rare it is that we drew a random sample and got our
estimate if the true value was really that specified in the
Null Hypothesis
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We can use software to help us
analyze the data
The production department of the Springfield Herald
newspaper has embarked on a quality improvement
effort and the first project is the blackness of the
newspaper print.
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Most often, we use software to
give us basic statistical data for
our analyses
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Blackness is measured on a standard scale in which
the target value is 1.0.
I want you to start your analysis
with basic descriptive statistics
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What do you see?
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A desirable level is that the average is at least .97.
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We want to determine if the average blackness is
different from .97, use alpha = .05
A random sample of 50 newspapers have been
selected
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The distribution looks
curiously bi-modal
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The mean is .959
Median is .985
Standard deviation is .156
CV = .156/.959*100 = 16.3
95% C.I. = .959 ± .044
= .915 to 1.003
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Hypothesis Test for the
Blackness of Newspaper Print
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If the sampling distribution for n=50 has a mean level of .970,
µ or the Hypothesized value
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I could find out how rare an event it was to take a sample of
50 and get a mean value of .959
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I can test to see how my sample compares to the
hypothesized population
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different from
Two-tailed
Greater than
One-Tailed Upper
or Less than
One-tailed Lower
Hypothesis Test
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We are going to use a rare-event approach to
make an inference from our sample to a population
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We define two hypotheses
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Null hypothesis – the hypothesis that will be
accepted unless we have convincing evidence
to the contrary - we seek to reject the Null
Hypothesis
Alternative Hypothesis – aka as the Research
Hypothesis. We look to see if the data provide
convincing evidence of its truth. We hope for
the alternative, but we are reluctant to accept
it outright.
Ha: µ ! .970
I will check if it is different
I calculate a z or t-score to see how far away it is
from the hypothesized value
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t* = (.959 – .970)/.022
SE = .156/(50).5 = .022
t* = -.011/.022
t* = -.500
Our sample estimate is .5 standard deviations
below the mean in relation to the sampling
distribution.
This is not so unusual.
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Here is our Strategy: Null and
Alternative Hypothesis
What is a Hypothesis Test?
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Ho: µ= .970
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The Null Hypothesis is based on expectations of no
change, nothing happening, no difference, the same old
same old
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In many ways it is a straw man (or person) and in
contrast to the rare event
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In most cases we want to reject the null hypothesis
The Alternative Hypothesis is more in line with our true
expectations for the experiment or research
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The sample value is not equal to the hypothesized value,
or
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It is greater than the hypothesized value, or
It is less than the hypothesized value
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The logic of a Hypothesis Test
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The logic of a Hypothesis Test
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We assume our sample estimate comes from a
population with a parameter as stated under the
null hypothesis
To do this we calculate a Test Statistic
This is a z-score (or a t-score) based on:
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We can compare our sample estimate to this
population parameter to see how likely it is that
our sample comes from a sampling distribution
from this population
Ho: µ = some value or
P = some value
The sample estimate of the standard deviation, s
The Standard Error of the sampling distribution for
our estimator, s/(n).5
The comparison is made via a Test Statistic
(x " µ)
!X
z* =
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z* =
( p " P)
#P
t* =
(x ! µ)
sX
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The components of a Hypothesis Test
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Ho:
Ha:
Assumptions
Test Statistic
Rejection Region
Calculation:
Conclusion:
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The components of a Hypothesis Test
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Ho: µ =.970
Ha: µ " .970
2-tailed
n= 50, # unknown, use t
t* = (.959 – .970)/.022
! = .05, .05/2, 49 d.f., t = ± 2.010
t* = -.50
t* > t.05/2, 49 df
-.50 > -2.010
Cannot Reject Ho: =.970
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Set up the Null Hypothesis
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Ho: µ = ???
Ho: µ = .970
some use " or # with a one-tailed test
Set up the Alternative Hypothesis
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It takes up one of three forms
Ha: ! .970
Two-tailed
Ha: > .970
One-tailed, upper tail
Ha: < .970
One-tailed, lower tail
Ha: " .970
Two-tailed
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The Alternative Hypothesis
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The Assumptions of the Test
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A one-tailed test of hypothesis is one in which the
alternative hypothesis is directional, and includes
either the < symbol or the > symbol.
Proportion
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Is the sample size large?
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A two-tailed test of hypothesis is one in which the
alternative hypothesis does not specify a particular
direction; it is “different.” It will be written with the "
symbol.
If Yes, use the normal
approximation to the binomial
Yes
z* =
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We gain by specifying a one-tailed test - more
knowledge is usually better in statistics
( p " P)
#P
If No, use the binomial
distribution
No
!
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The Assumptions of the Test
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! The Test Statistic
Mean
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Is sigma known?
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Yes
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No
If Yes, use the known ! for the
standard error and use z
If No, use the sample estimate,
s, for the standard error, and
use t
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Yes
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No
If n is small, especially if ! is
unknown, we have to worry
about whether the population
is distributed as normal
Let t* equal the calculated test statistic
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If we don’t know $ we use the sample
estimate of s to calculate the standard error
(.959 ! .970)
as (s/n.5)
t* =
= !.50
.156
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This t* represents a “z-score” of our sample
compared to the sampling distribution of the
mean, if the null hypothesis is true
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(x ! µ)
sX
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Is sample size small?
t* =
The value for µ will come from the Null
Hypothesis
We want to see how far away our sample
estimate is from the null hypothesis
population mean
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The Critical Value and the
Rejection Region
The Critical Value and the
Rejection Region
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We look to see where t* falls in relation to the tdistribution
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If it lies in the Rejection Region, in one of the tails,
it is possible that it came from that hypothesized
distribution -- but not very likely
Alpha (%) is the probability level at which you are willing
to be wrong when rejecting Ho.
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The value at the boundary of the rejection region is
called the Critical Value
Because it is unlikely, I am willing to reject the Null
Hypothesis
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We establish the Critical Value linked to ! and the
type of test (1 or 2-tailed) and compare our Test
Statistic to this value
We need to set the level of alpha and the type of
test (1 or 2-tailed) a priori
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The rejection region is the set of possible values of the
test statistic for which the null hypothesis will be
rejected.
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So if the calculated test statistic falls within the rejection
region, we reject Ho
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More on the Rejection Region
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I want to pick an alpha value far out in the tail of
the sampling distribution
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So if I find a difference, I can be reasonably sure
that my sample is not part of the sampling
distribution specified under the null hypothesis
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% represents the probability that my sample mean
actually comes from the hypothesized sampling
distribution, but I’m going to say it doesn’t
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More on the Rejection Region
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For this problem, %=.05
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This is a two-tailed test as specified in the alternative
hypothesis
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This is called a Type I Error
The probability of committing a Type I Error is
the probability of rejecting Ho when Ho is true
Next I want to find a t-value that will correspond with an
overall % =.05, with n-1 degrees of freedom
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Ha: µ ! .970
So if % = .05, we need to split this probability level in
the upper and lower tail of the sampling distribution,
much like we do for a confidence interval
If it were a one-tailed test, we would put all of alpha into
one tail
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Rejection Region
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So I look for a t-score that corresponds to an
overall probability of .05
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In the t-table this is a value relating to
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Look at it in pictures
Our test statistic is t* = -.50
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If our test statistic is in the
Rejection Region, we would
consider it far enough away
from the Null Value
% = .05 for two-tails, or .025 in one tail
d.f. = 49
The t-value from the table is 2.010
The values of -2.010 and 2.010 are the critical
values which marks the rejection region
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So our Rejection Region is:
Any test statistic value < –2.010 or > 2.010
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To reject the Null Hypothesis
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Otherwise, we Fail to Reject
So what do I conclude?
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I don’t have any evidence that the average
blackness of the print is different from .97.
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It was not so unusual to draw a random sample of
50 papers and get an average of .959 given the
true value is .97
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So I can’t conclude I have a problem from this
sample…
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….but I could be wrong!!!
The Critical Values that mark
the rejection region are -2.010
and 2.010
t*=-.50
-2.010
Which makes the Alternative
Hypothesis more plausible
2.010
My test statistic does not fall in the
rejection region, therefore I cannot
reject the Null Hypothesis
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The components of a Hypothesis Test
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Ho:
Ha:
Assumptions
Test Statistic
Rejection Region
Calculation:
Conclusion:
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Ho: µ =.970
Ha: µ " .970
2-tailed
n= 50, # unknown, use t
t* = (.959 – .970)/.022
! = .05, .05/2, 49 d.f., t = ± 2.010
t* = -.50
t* > t.05/2. 49 df
-.50 > -2.010
Cannot Reject Ho: =.970
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What if this were a one-tailed
test?
Hypothesis Test from JMP
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Here is the output from JMP
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You would be asked to specify a
Hypothesized Value, Ho
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An alpha level, .05
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Then it gives the results of a one
or two-tailed test
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What if we multiplied blackness
by 100?
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Summary
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We established the basis for a Hypothesis Test
We base it on a Rare Event Approach
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We ask how rare an event it was to draw a sample of
size n
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and observe our sample estimate - mean or p
If the the true value were really that specified in the Null
Hypothesis
To do this we need
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A Null and Alternative Hypothesis
A Test Statistic, z* or t*
A criterion for decision making - how far away is enough
to reject the Null Hypothesis - based on !
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Let’s say that the Alternative
Hypothesis was:
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Ha: µ < .97
% = .05
The test statistic remains the
same t* = -.50
Now, all of alpha would fall in
just one tail
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t*=-.50
t.05, 49 df = -1.677
This shifted the Rejection
Region to the right and made it
easier to reject the Null
Hypothesis
-1.677
My test statistic does not fall in the
rejection region, therefore I cannot
reject the Null Hypothesis
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